Method for determining the humidity and density of a dielectric material

ABSTRACT

The invention relates to a method for determining the humidity and/or density of a dielectric material in a resonator that is filled with said material and that contains a transmitter and a receiver. According to said method: the transmitter emits a signal; a resonance curve of the filled resonator is scanned in stages, whereby respective signal intensity values (U i ) are measured at different frequencies (f i ); the resonant frequency (f rm ) and the bandwidth (BW m ) are determined for the filled resonator from measured points (f i /U i ); and the humidity (ψ) and/or density (ρ) of the material are calculated by solving a second system of equations (G2), containing the resonant frequencies (f r0 , f rm ) and bandwidths (BW 0 , BW m ) of the empty and filled resonators and known calibration coefficients (a r1 , a r2 , b r1 , b r2 , c r1 , c r2 , a bw1 , a be2 , b bw1 , b bw2 , c bw1 , c bw2 ) of said resonator. The aim of the invention is to provide a method for determining the humidity independently of the density in a rapid, precise manner.

BACKGROUND OF THE INVENTION

The invention relates to a method for determining the humidity and/ordensity of a dielectric material in a resonator filled with thematerial, comprising a sender and a receiver.

The dielectric properties of a material, described by the complexrelative dielectric constant ∈_(r)=∈′_(r)−i∈″_(r), can be affected byhumidity and density in case of porous materials. Humidity and densitythus modify the scalar parameters resonant frequency f_(r) and resonatorquality Q of a material-filled resonator in contrast to those of theempty, air-filled resonator (f_(r) ₀ and Q₀) in such a way that

${f_{r_{m}} = {{\frac{f_{r_{0}}}{\sqrt{ɛ_{r}}}\mspace{14mu}{and}\mspace{14mu}\frac{1}{Q_{m}}} = {\frac{1}{Q_{0}} + \frac{ɛ_{r}^{''}}{ɛ_{r}^{\prime}}}}},$wherein 1≦∈′_(r)≦∈′_(r) _(max) and 0≦∈″_(r)≦∈″_(r) _(max) , wherein∈′_(r) _(max) and ∈″_(r) _(max) are the maximum values resulting for therespective material from the assigned humidity range and density range.

In prior art, different methods are known for measuring the humidity ordensity of granular materials, in which the resonance behavior of amatter-filled resonator is used.

For example, from U.S. Pat. No. 5,666,061 a method is known formeasuring the humidity in granular materials by means of microwaves.There, one electronically reacts to a threshold while bidirectionallysweeping a frequency range. From the chronological properties of pulsescreated in such a way the components of the dielectric constant arededuced.

All methods are relatively slow and have either a dependency of thedensity of the material or have relatively large errors in thedetermination of the humidity.

The invention is underlied by the problem to specify a method of thetype initially mentioned, by which a fast, accurate anddensity-independent determination of the humidity is possible.

SUMMARY OF THE INVENTION

For the purpose of the invention, any arbitrary quantity describing theresonance width can be considered as a bandwidth in the following,wherein appropriate adjustments have to be provided in the respectivedefinitions, in particular of the threshold values, and equations.

By digitally recording the resonance curve, a fast acquisition of themeasuring values is possible. Thus, a close chronological-spatialassignment of the humidity determined from these values to materialdynamically guided through the resonator is possible. For this purpose,the invention provides for that the sender emits a signal; a resonancecurve of the filled resonator is sweeped, wherein respective relatedsignal strength values of the receiver signal are measured at differentfrequencies; the resonant frequency and the bandwidth are determined forthe filled resonator from the measured points; and the humidity and/ordensity of the material is calculated by solving a second system ofequations comprising the resonant frequencies and the bandwidths of theempty and of the filled resonator and known calibration coefficients ofthe resonator. This method enables a density-independent determinationof the humidity of the material. Besides, the density of the materialcan be acquired with little effort.

A preferred embodiment provides for that, from the points fordetermining the bandwidth of the filled resonator, either the quantitiesresonant frequency, resonator quality and resonance maximum aredetermined and the bandwidth is calculated therefrom, or cut-offfrequencies are determined and the resonant frequency and the bandwidthare calculated therefrom.

In another embodiment a lower threshold value is calculated and a secondsweeping pass with smaller step sizes is performed in that range inwhich the signal strength values are higher than the threshold value. Bya two-pass procedure the accuracy can be significantly increased and,nevertheless, the required time can be kept short, in particular if,during the second sweeping pass, a decreased step size is used in arange of the resonance peak only.

Advantageously, sweeping the resonance curve is performed in equallyspaced steps. The simplest and fastest form of sweeping consists ofequally spaced steps. Variable step sizes in the second sweeping passcan shorten its duration.

Preferably, the sender is operated using a constant strength. Themeasuring values at the receiver can be used without adjusting orscaling them in case of a constant signal strength.

In a possible embodiment, the cut-off frequencies of the resonator aredetermined by determining the point having the highest receiver signalstrength value, and, starting from this point, calculating a thresholdvalue; determining two respective proximate points for positive andnegative slope sections, the signal strength values of these pointslying below and above the threshold value; calculating first and secondcut-off frequencies therefrom by respectively interpolating between theproximate points. Determining the cut-off frequencies by interpolatingbetween point pairs surrounding a threshold value as an initialparameter for determining the humidity and/or density is a fast andsimple method.

An advantageous embodiment thereby provides for that the threshold valuecorresponds to an attenuation of 3 dB in relation to the highest signalvalue. If the threshold value is chosen corresponding to an attenuationof 3 dB, starting from the maximum signal strength value, the equationsto be solved obtain a very simple form.

In an alternative embodiment the quantities resonant frequency,resonator quality and resonance maximum of the resonator are determinedby arbitrarily and/or randomly selecting three points and solving afirst system of equations for these quantities, the system consisting ofthree equations of an analytic resonance curve that are valid for thethree points. The resonant frequency, the resonator quality and theresonance amplitude can be directly determined as the initial parametersfor determining the humidity and/or density even faster and with lesserror than by interpolation by solving a first system of equationsconsisting of three equations and thus being completely determined,wherein for each point out of a group of three one of the threeequations is valid, the three points being selected from the presentset. In particular, noisy resonance curves can be analyzed more exactlythis way.

In another alternative embodiment the quantities resonant frequency,resonator quality, resonance maximum of the resonator are determined byarbitrarily and/or randomly selecting a set of points whose number is aninteger multiple of three and at least six, and splitting up the pointset into three equally sized groups; for each combination of threepoints, wherein each point comes from a different group, solving a firstsystem of equations for these quantities, the system consisting of threeequations of the analytic resonance curve valid for these three points;and creating the average for each quantity from the values calculated atthe combinations. Even more exact values than with three points can beobtained by creating several triple-groups from the present point setand averaging the initial parameters obtained thereby over all groups.

In an advantageous embodiment, as a condition for arbitrarily and/orrandomly selecting the points, the signal value of a point to beselected is higher than the highest signal value attenuated by 3 dB. Inboth procedures using three points or a multiple thereof, preferablyonly those points from the resonance curve are selected whose signalstrength is higher than the maximum value of all measurement values,attenuated by 3 dB, as this way only significant values are used.

The system of equations to be solved is preferably chosen in such a waythat it describes, in a good approximation, the correlation of humidityand density with a variation of resonant frequency and resonator qualityor, respectively, with a variation of resonant frequency and bandwidth.

Advantageously the second system of equations is non-linear.

In a preferred embodiment the sweeping by means of the sender isperformed up to the microwave area.

Advantageously, voltage values or current values of the receiver areused for measuring the receiver signal. Preferably, the electricalvoltage rising at the receiver serves for acquiring the resonance curve,because it is measurable easily and without back coupling. However, thecurrent in a receiver circuit can be measured, too.

In the following, the denomination U_(i) is used for the measuredquantities. However, in doing so, it shall not be implied to use thevoltage only.

The resonance curve to be reconstructed from the discrete measuringpoints is an approximation as-good-as-possible to the real resonancecurve only. It is characterized by the course of the signal amplitude Uat the receiver against the supplied frequency f, wherein U_(r) is theresonance maximum and Q is the quality of the resonance:

$U = \frac{U_{r}}{\sqrt{1 + {Q^{2}\left( {\frac{f}{f_{r}} - \frac{f_{r}}{f}} \right)}^{2}}}$

The cut-off frequencies f_(a) and f_(b) are those frequencies at whichthe signal strength exceeds and falls below a defined value,respectively. For this purpose, preferably the value corresponding to anattenuation of the maximum by 3 dB is chosen: U_(a)=U_(b)=U_(r)/√{squareroot over (2)}. Between the cut-off frequencies and the curve parametersresonant frequency f_(r) and resonator quality Q there are thecorrelations:

${f_{r} = \sqrt{f_{a} \cdot f_{b}}},{Q = {\frac{\sqrt{f_{a} \cdot f_{b}}}{f_{b} - f_{a}}.}}$The distance between the cut-off frequencies is defined as the bandwidth

${BW} = {{f_{b} - f_{a}} = {\frac{f_{r}}{Q}.}}$

In the following, the invention is described in further detail usingexamples of embodiments.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows a sweeped resonance curve;

FIG. 2 shows a resonance curve sweeped in a two-pass procedure;

FIG. 3 shows a first way for determining the resonance parameters;

FIG. 4 shows a second way for determining the resonance parameters;

FIG. 5 shows a third way for determining the resonance parameters; and

FIG. 6 shows a two-dimensional representation of calibration curves.

DETAILED DESCRIPTION OF THE INVENTION

In FIG. 1, two resonance curves are depicted above each other like theyhave been recorded using the method according to the invention; theright resonance curve at an empty, the left resonance curve at amaterial-filled resonator.

By sweeping (wobbling) the resonator between the start frequencyf_(start1) and the stop frequency f_(stop1), the resonance curve isacquired in discrete steps. The start frequency is calculated from themaximally shifted frequency

${f_{r_{m}} = \frac{f_{r_{0}}}{\sqrt{ɛ_{r_{\max}}}}},$the maximally changed quality

$\frac{1}{Q_{m}} = {\frac{1}{Q_{0}} + ɛ_{r_{\max}}^{''}}$and the normalized voltage ratio

$a = {{\frac{U_{a}}{U_{\max}} < {\frac{1}{\sqrt{2}}\text{:}f_{{start}\; 1}}} = {{{- \frac{f_{r_{m}}}{2Q_{m}}}\sqrt{\frac{1 - a^{2}}{a^{2}}}} + {\sqrt{{\left( \frac{f_{r_{m}}}{2Q_{m}} \right)^{2} \cdot \frac{1 - a^{2}}{a^{2}}} + f_{r_{m}}^{2}}.}}}$The stop frequency, using f_(r)=f_(r) ₀ , results as:

$f_{{stop}\; 1} = {{{- \frac{f_{r\; 0}}{2Q_{m}}}\sqrt{\frac{1 - a^{2}}{a^{2}}}} + {\sqrt{{\left( \frac{f_{r\; 0}}{2Q_{m}} \right)^{2} \cdot \frac{1 - a^{2}}{a^{2}}} + f_{r_{0}}^{2}}.}}$

The sweeping speed depends, among others, on the number n₁ of sweepingpoints and can be increased by a two-pass procedure as depicted in FIG.2. For this purpose, first a sweeping is performed using a smallernumber of sweeping points and, hence, a larger frequency step size

${\Delta\; f_{1}} = \frac{f_{{stop}\; 1} - f_{{start}\; 1}}{n_{1} - 1}$and, in a second iteration, after previously determining the start andstop frequencies f_(start2)=f|U≈U_(a Λf >0), f_(stop2)=f|U≈U_(a Λf <0)or f_(start2)=f|U≈U_(a ΛU<U) _(r), f_(stop2)=f|_(U≈U) _(a)ΛU>Ur, anothersweeping is subsequently performed using a smaller frequency step size

${\Delta\; f_{2}} = \frac{f_{{stop}\; 2} - f_{{start}\; 2}}{n_{2} - 1}$between these frequencies.

From the sweeped, measured resonance curve the resonator parametersf_(r), Q and U_(r) can be determined, for example, according to the fafbprocedure, the three-points procedure or the 3k-points procedure. Otherprocedures are possible, too.

FIG. 3 shows the fafb procedure. It is based on directly determining afirst and a second cut-off frequency f_(a), f_(b) from the measuredresonance curve. For this purpose, first the point (also called sweepingpoint) (f=f_(max)/U=U_(max))having the highest voltage U=U_(max) isdetermined for calculating a 3 dB threshold line

$U_{3d\; B} = {\frac{U_{\max\;}}{\sqrt{2}}.}$Subsequently, the respective two points in the immediate proximity ofthe 3 dB threshold line are determined. By linearly interpolatingbetween the sweeping points a1 and a2 as well as b1 and b2 the first andthe second 3 dB cut-off frequency f_(a), f_(b) are obtained:

${f_{a} = {f_{a\; 1} + {\frac{\frac{U_{\max}}{\sqrt{2}} - U_{a\; 1}}{U_{a\; 2} - U_{a\; 1}}\left( {f_{a\; 2} - f_{a\; 1}} \right)}}},{f_{b} = {f_{b\; 1} + {\frac{\frac{U_{\max}}{\sqrt{2}} - U_{b\; 1}}{U_{b\; 2} - U_{b\; 1}}{\left( {f_{b\; 2} - f_{b\; 1}} \right).}}}}$From the equations

${f_{r} = \sqrt{f_{a} \cdot f_{b}}},{Q = \frac{\sqrt{f_{a} \cdot f_{b}}}{f_{b} - f_{a}}},$f_(r) and Q can be calculated. The resonance voltage U_(r) then resultsas:

$U_{r} = {U_{\max} \cdot {\sqrt{1 + {Q^{2}\left( {\frac{f_{\max}}{f_{r}} - \frac{f_{r}}{f_{\max}}} \right)}^{2}}.}}$

Calculating the cut-off frequencies is more or less error-prone becauseof the discretization of the resonance curve—on one hand, by thedetermination of the 3 dB threshold line (U_(a)=U_(max)/√{square rootover (2)}≦U_(r)/√{square root over (2)}) derived from U_(max) and, onthe other hand, by the linear interpolation between the sweeping pointsa1, a2, b1, b2. A higher sweeping rate indeed reduces the errors, butalso increases the sweeping time and thus reduces the sweeping speed. Atbest, a non-linear interpolation is possible in this procedure in orderto improve accuracy.

In the three-points procedure shown in FIG. 4, three sweeping points U₁,U₂, U₃ are selected arbitrarily or by a random generator preferablyabove the line U_(s)=s·U_(max) with

$s < {\frac{1}{\sqrt{2}}.}$

By solving the first system of equations G1:

${U_{1} = \frac{U_{r}}{\sqrt{1 + {Q^{2}\left( {\frac{f_{1}}{f_{r}} - \frac{f_{r}}{f_{1}}} \right)}^{2}}}},{U_{2} = \frac{U_{2}}{\sqrt{1 + {Q^{2}\left( {\frac{f_{2}}{f_{r}} - \frac{f_{r}}{f_{2}}} \right)}^{2}}}},{U_{3} = \frac{U_{3}}{\sqrt{1 + {Q^{2}\left( {\frac{f_{3}}{f_{r}} - \frac{f_{r}}{f_{3}}} \right)}^{2}}}}$the resonator parameters f_(r), Q and U_(r) are obtained:

$f_{r} = \sqrt[4]{\frac{{{U_{1}^{2}\left( {U_{3}^{2} - U_{2}^{2}} \right)}f_{1}^{2}} + {{U_{2}^{2}\left( {U_{1}^{2} - U_{3}^{2}} \right)}f_{2}^{2}} + {{U_{3}^{2}\left( {U_{2}^{2} - U_{1}^{2}} \right)}f_{3}^{2}}}{\frac{U_{1}^{2}\left( {U_{2}^{2} - U_{3}^{2}} \right)}{f_{1}^{2}} + \frac{U_{2}^{2}\left( {U_{3}^{2} - U_{1}^{2}} \right)}{f_{2}^{2}} + \frac{U_{3}^{2}\left( {U_{1}^{2} - U_{2}^{2}} \right)}{f_{3}^{2}}}}$$Q = \sqrt{\frac{U_{1}^{2} - U_{2}^{2}}{{U_{2}^{2}\left( {\frac{f_{2}}{f_{r}} - \frac{f_{r}}{f_{2}}} \right)}^{2} - {U_{1}^{2}\left( {\frac{f_{1}}{f_{r}} - \frac{f_{r}}{f_{1}}} \right)}^{2}}}$$U_{r} = {U_{1}\sqrt{1 + {Q^{2}\left( {\frac{f_{1}}{f_{r}} - \frac{f_{r}}{f_{1}}} \right)}^{2}}}$

The advantage of the three-points procedure in relation to the fafbprocedure lies in that no errors occur because of the discretization ofthe resonance curve, and that only few points are needed for sweepingthe resonance curve, and thus a high sweeping speed is achieved.

If the resonance curve is noisy, error-prone resonator parameters aredetermined by the three-points procedure, too. In order to reduce theimpact of the noise, more than three sweeping points are selected,namely a number3·k, k=2, 3, 4, . . ., and are split up into three groupsM1, M2, M3, having k points per group above the line U_(s)=s·U_(max).FIG. 5 shows this 3k-points procedure. For all k³ possibilities ofcombining one respective point from each of the three groups, theresonator parameters are determined from the first system of equationsG1 according to the three-points procedure described above, and the k³partial results are averaged subsequently:

${f_{r} = {\frac{1}{k^{3}}{\sum\limits_{j = 1}^{k^{3}}f_{r_{j}}}}},{Q = {\frac{1}{k^{3}}{\sum\limits_{j = 1}^{k^{3}}Q_{j}}}},{U_{r} = {\frac{1}{k^{3}}{\sum\limits_{j = 1}^{k^{3}}{U_{r_{j}}.}}}}$

In order to determine humidity and density from determined resonatorparameters, the used arrangement is calibrated using material of knownhumidity and density. The calibration is advantageously performed out ofthe regular operation.

In the calibration, the electrical resonator values resonant frequencyf_(r), resonator quality Q and bandwidth BW(BW=f_(r)/Q) are respectivelyassigned to the material quantities humidity ψ and density ρ. Thisassignment between the material values and the electrical values isnon-linear and can be described for a defined range of humidity anddensity with sufficient accuracy, for example, by the following secondsystem of equations G2:Δf _(r) =a _(f) _(r2) ρ²ψ² +a _(f) _(r1) ρψ² +b _(f) _(r2) ρ² ψ+b _(f)_(r1) ρψ+c _(f) _(r2) ρ² +c _(f) _(r1) ρΔBW=a _(BW) ₂ ρ²ψ² +a _(BW) ₁ ρψ² +b _(BW) ₂ ρ² ψ+b _(BW) ₁ ρψ+c _(BW) ₂ρ² +c _(BW) ₁ ρ

Δf_(r) and ΔBW are the differences of resonant frequency and bandwidth,respectively, between the empty and the material-filled resonator:Δf _(r) =f _(r) ₀ −f _(r) _(m) , ΔBW=BW _(m) −BW ₀

Now, it is an object of the calibration to determine the twelvecalibration coefficients a_(f) _(r1) , a_(f) _(r2) , b_(f) _(r1) , b_(f)_(r2) , c_(f) _(r1) , c_(f) _(r2), a_(BW) ₂ , a_(BW) ₁ , b_(BW) ₁ ,b_(BW) ₂ , c_(BW) ₁ , c_(BW) ₂ for the respective material from asufficient large number of calibration values (Δf_(r) and ΔBW withrelated material values ψ und ρ).

For this purpose, the measuring values of the resonant frequencydifference Δf_(r) and the bandwidth difference ΔBW are assigned to thehumidity and density values that have been determined by appropriatereference methods. These calibration values characterizing the resonantfrequency and bandwidth depending on humidity and density are the basisfor determining the calibration coefficients. The regression of thecalibration values of equal humidity is performed in the form ofΔf_(r)=d_(f) _(r) ₂ρ²+d_(f) _(r) ₁ρ and Δf_(r)=d_(BW) ₂ ρ²+d_(BW) ₁ ρ,wherein the regression curves have to run through the origin, as Δf_(r)and ΔBW are zero for the empty resonator (air having ρ=0). Theregression provides calibration points of equal density depending on thehumidity. Using these, another regression of the form Δf_(r)=a_(f) _(r)ψ²+b_(f) _(r) ψ+c_(f) _(r) and ΔBW=a_(BW)ψ²+b_(BW)ψ+c_(BW) is performed.The values determined this way for the regression coefficients a_(f)_(r) , b_(f) _(r) , c_(f) _(r) and a_(BW), b_(BW), c_(BW) are plottedagainst the density and, therefrom, the calibration coefficients aredetermined by a quadratic regression.

FIG. 6 shows the results of the calibration in the form of a set ofcurves which also serves for analysis.

From the measured resonant frequency difference values Δf_(r) andbandwidth difference values ΔBW humidity content ψ and density ρ arecalculated for the respective material by solving the above secondsystem of equations G2.

In doing so, two real and two imaginary roots result. From the course ofthe calibration curves in the Δf_(r-)ΔBW diagram it can be determined ifthere is only one real solution in the humidity and density range ofinterest. In the Δf_(r-)ΔBW diagram the bandwidth difference isrepresented in dependency of the resonant frequency difference forcurves of equal density and humidity. If the course of these curves,characterized by the points A, B, C und D in FIG. 6, is continuous andunique in the humidity and density range of interest then only one realsolution exists in this range.

For solving the second system of equations G2 an iterative procedure isappropriate. For this purpose, the second system of equations G2 issolved for ψ:

$\psi = {{- \frac{b_{f_{r}2\varrho} + b_{f_{r}1}}{{2a_{f_{r}2\varrho}} + {2a_{f_{r}1\varrho}}}} + \sqrt{\left( \frac{b_{f_{r}2\varrho} + b_{f_{r}1}}{{2a_{f_{r}2\varrho}} + {2a_{f_{r}1\varrho}}} \right)^{2} - \frac{c_{f_{r}2\varrho^{2}} + c_{f_{r}1} - {\Delta\; f_{r_{0}}}}{a_{f_{r}2\varrho^{2}} + a_{f_{r}1\varrho}}}}$$\psi = {{- \frac{b_{{BW}_{2}\varrho} + b_{{BW}_{1}}}{{2a_{{BW}_{2}\varrho}} + {2a_{{BW}_{1}\varrho}}}} + \sqrt{\left( \frac{b_{{BW}_{2}\varrho} + b_{{BW}_{1}}}{{2a_{{BW}_{2}\varrho}} + {2a_{{BW}_{1}\varrho}}} \right)^{2} - \frac{c_{{BW}_{2}\varrho^{2}} + c_{{BW}_{1}} - {\Delta\;{BW}_{0}}}{a_{{BW}_{2}\varrho^{2}} + a_{{BW}_{1}\varrho}}}}$

From the intersection of both equations in a humidity-density diagramthe sought values for ψ and ρ are obtained.

LIST OF REFERENCE NUMERALS

-   f_(r) Resonant frequency in the general case-   f_(r) ₀ Resonant frequency of the empty resonator-   f_(r) _(m) Resonant frequency of the filled resonator-   f_(start1), f_(stop1) Start and stop frequencies for first sweeping    pass-   U_(max) Highest signal strength value measured-   a, s Threshold value factors-   f_(start2), f_(stop2) Start and stop frequencies for second sweeping    pass-   f_(max) Frequency at which the highest signal strength value is    present-   (f_(a1)/U_(a1)), (f_(a2)/U_(a2)) Proximate points of the first    cut-off frequency-   (f_(b1)/U_(b1)), (f_(b2)/U_(b2)) Proximate points of the second    cut-off frequency-   (f₁/U₁) . . . (f₃/U₃) Three selected points-   M1, M2, M3 point groups-   (f₁₁/U₁₁) . . . (f₁₄/U₁₄) Elements of point group M1-   (f₂₁/U₂₁) . . . (f₂₄/U₂₄) Elements of point group M2-   (f₃₁/U₃₁) . . . (f₃₄/U₃₄) Elements of point group M3-   A,B,C,D Limits of the humidity and density range of interest

1. A method for determining the humidity and/or density of a dielectricmaterial in a resonator filled with the material, the resonatorincluding a sender and a receiver, the method comprising: emitting asignal by the sender; sweeping a resonance curve of the filledresonator; measuring appropriate signal strength values of the receiversignal at respective different frequencies; determining a resonantfrequency and a bandwidth for the filled resonator from pointscorresponding to the signal strength values of the receiver signal atthe respective different frequencies measured; and calculating at leastone of humidity or density of the material by solving a second system ofequations comprising the resonant frequencies and respective bandwidthsof the empty and of the filled resonator and known calibrationcoefficients of the resonator, wherein, from the points for determiningthe bandwidth of the filled resonator, cut-off frequencies aredetermined and the resonant frequency and the bandwidth are calculatedtherefrom, and wherein the cut-off frequencies of the resonator aredetermined by: determining a one of the points having a highest signalstrength value, and, starting from said one of the points, calculating athreshold value; determining two proximate points for positive andnegative slope sections, the signal values of said two proximate pointslying below and above the threshold value, respectively; and calculatingfirst and second cut-off frequencies therefrom by respectivelyinterpolating between the two proximate points.
 2. The method accordingto claim 1, wherein the threshold value corresponds to an attenuation of3 dB in relation to the highest signal strength value.
 3. A method fordetermining the humidity and/or density of a dielectric material in aresonator filled with the material, the resonator including a sender anda receiver, the method comprising: emitting a signal by the sender;sweeping a resonance curve of the filled resonator; measuringappropriate signal strength values of the receiver signal at respectivedifferent frequencies; determining a resonant frequency and a bandwidthfor the filled resonator from points corresponding to the signalstrength values of the receiver signal at the respective differentfrequencies measured; and calculating at least one of humidity ordensity of the material by solving a second system of equationscomprising the resonant frequencies and respective bandwidths of theempty and of the filled resonator and known calibration coefficients ofthe resonator, wherein, from the points for determining the bandwidth ofthe filled resonator, the quantities resonant frequency, resonatorquality and resonance maximum are determined and the bandwidth iscalculated therefrom, and wherein the quantities resonant frequency,resonator quality, and resonance maximum of the resonator are determinedby: at least one of arbitrarily or randomly selecting a set of thepoints for which a number is an integer multiple of three and is atleast six, and splitting up the point set into three equally sizedgroups; for each combination of three points, wherein each point comesfrom a different one of the groups, solving a first system of equationsto obtain resonant parameters, the system consisting of three equationsof the analytic resonance curve valid for said three points; and foreach of the resonant parameters, creating an average of valuescalculated at said combinations.
 4. The method according to claim 3,wherein, as a condition for the at least one of arbitrarily or randomlyselecting the points, the signal value of a particular one of the pointsto be selected is higher than the highest signal value attenuated by 3dB.
 5. The method according to claim 1 or 3, wherein the second systemof equations describes, as at least an approximation, the correlation ofhumidity and density with the variation of resonant frequency andresonator quality, or with the variation of resonant frequency andbandwidth, in a predefined range of humidity and density.
 6. The methodaccording to claim 1 or 3, wherein the second system of equations isnon-linear.
 7. The method according to claim 1 or 3, wherein thesweeping by the sender is performed up to the microwave area.
 8. Themethod according to claim 1 or 3, wherein voltage values or currentvalues of the receiver are used for measuring the receiver signal.